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volume 3, issue 4, article 49, 2002.

Received 04 April, 2002;

accepted 01 May, 2002.

Communicated by:A. Fiorenza

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Journal of Inequalities in Pure and Applied Mathematics

SHARP ERROR BOUNDS FOR THE TRAPEZOIDAL RULE AND SIMP- SON’S RULE

D. CRUZ-URIBE AND C.J. NEUGEBAUER

Department of Mathematics Trinity College

Hartford, CT 06106-3100, USA.

EMail:david.cruzuribe@mail.trincoll.edu Department of Mathematics

Purdue University West Lafayette, IN 47907-1395, USA.

EMail:neug@math.purdue.edu

c

2000School of Communications and Informatics,Victoria University of Technology ISSN (electronic): 1443-5756

031-02

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Sharp Error Bounds for the Trapezoidal Rule and Simpson’s

Rule

D. Cruz-Uribe,C.J. Neugebauer

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J. Ineq. Pure and Appl. Math. 3(4) Art. 49, 2002

http://jipam.vu.edu.au

Abstract

We give error bounds for the trapezoidal rule and Simpson’s rule for “rough”

continuous functions—for instance, functions which are Hölder continuous, of bounded variation, or which are absolutely continuous and whose derivative is in L^p. These differ considerably from the classical results, which require the functions to have continuous higher derivatives. Further, we show that our results are sharp, and in many cases precisely characterize the functions for which equality holds. One consequence of these results is that for rough functions, the error estimates for the trapezoidal rule are better (that is, have smaller constants) than those for Simpson’s rule.

2000 Mathematics Subject Classification:26A42, 41A55.

Key words: Numerical integration, Trapezoidal rule, Simpson’s rule

1. Introduction

1.1. Overview of the Problem

Given a finite intervalI = [a, b]and a continuous functionf :I →R, there are two elementary methods for approximating the integral

Z

I

f(x)dx,

the trapezoidal rule and Simpson’s rule. Partition the intervalI intonintervals of equal length with endpointsx_i =a+i|I|/n,0≤i≤n. Then the trapezoidal rule approximates the integral with the sum

(1.1) T_n(f) = |I|

2n f(x₀) + 2f(x₁) +· · ·+ 2f(xn−1) +f(x_n) .

Similarly, if we partition I into2n intervals, Simpson’s rule approximates the integral with the sum

(1.2) S2n(f) = |I|

6n f(x0) + 4f(x1) + 2f(x2) + 4f(x3) +· · ·

+ 4f(x2n−1) +f(x_2n) . Both approximation methods have well-known error bounds in terms of higher derivatives:

E_n^T(f) =

T_n(f)− Z

I

f(x)dx

≤ |I|³kf⁰⁰k∞

12n² , E_2n^S (f) =

S2n(f)− Z

I

f(x)dx

≤ |I|⁵kf⁽⁴⁾k_∞ 180n⁴ .

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(See, for example, Ralston [13].)

Typically, these estimates are derived using polynomial approximation, which leads naturally to the higher derivatives on the righthand sides. However, the assumption thatf is not only continuous but has continuous higher order derivatives means that we cannot use them to estimate directly the error when approximating the integral of such a well-behaved function asf(x) =√

xon[0,1]. (It is possible to use them indirectly by approximatingf with a smooth function;

see, for example, Davis and Rabinowitz [3].)

In this paper we consider the problem of approximating the errorE_n^T(f)and E_2n^S (f)for continuous functions which are much rougher. We prove estimates of the form

(1.3) E_n^T(f), E_2n^S (f)≤c_nkfk;

where the constants cn are independent of f, cn → 0 as n → ∞, and k · k denotes the norm in one of several Banach function spaces which are embedded inC(I). In particular, in order (roughly) of increasing smoothness, we consider functions in the following spaces:

• Λ_α(I),0< α≤1: Hölder continuous functions with norm

kfk_Λ_α = sup

x,y∈I

|f(x)−f(y)|

|x−y|^α .

• CBV(I): continuous functions of bounded variation, with norm

kfk_BV,I = sup

Γ n

X

i=1

|f(x_i)−f(x_i−1)|,

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wherea =x₀ < x₁ < · · ·< x_n = b, and the supremum is taken over all such partitionsΓ ={xi}ofI.

• W₁^p(I),1≤p≤ ∞: absolutely continuous functions such thatf⁰ ∈L^p(I), with normkf⁰k_p,I.

• W₁^pq(I), 1 ≤ p ≤ ∞: absolutely continuous functions such that f⁰ is in the Lorentz spaceL^pq(I), with norm

||f⁰||_pq,I = Z ∞

0

t^q/p−1(f⁰)^∗q(t)dt 1/q

= p

q Z ∞

0

λ_f⁰(y)^q/pd(y^q) 1/q

. (For precise definitions, see the proof of Theorem1.6in Section4below, or see Stein and Weiss [17].)

• W₂^p(I), 1 ≤ p ≤ ∞: differentiable functions such that f⁰ is absolutely continuous andf⁰⁰ ∈L^p(I), with normkf⁰⁰k_p,I.

(Properly speaking, some of these norms are in fact semi-norms. For our purposes we will ignore this distinction.)

In order to prove inequalities like (1.3), it is necessary to make some kind of smoothness assumption, since the supremum norm on C(I)is not adequate to produce this kind of estimate. For example, consider the family of functions {f_n}defined on[0,1]as follows: on[0,1/n]let the graph off_nbe the trapezoid with vertices(0,1),(1/n²,0),(1/n−1/n²,0),(1/n,1), and extend periodically with period1/n. ThenE_n(f_n) = 1−1/nbut||f_n||∞,I = 1.

Our proofs generally rely on two simple techniques, albeit applied in a some- times clever fashion: integration by parts and elementary inequalities. The idea

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of applying integration by parts to this problem is not new, and seems to date back to von Mises [19] and before him to Peano [11]. (This is described in the introduction to Ghizzetti and Ossicini [7].) But our results themselves are either new or long-forgotten. After searching the literature, we found the following papers which contain related results, though often with more difficult proofs and weaker bounds: Pólya and Szegö [12], Stroud [18], Rozema [15], Rahman and Schmeisser [14], Büttgenbach et al. [2], and Dragomir [5]. Also, as the final draft of this paper was being prepared we learned that Dragomir, et al. [4] had independently discovered some of the same results with similar proofs. (We would like to thank A. Fiorenza for calling our attention to this paper.)

1.2. Statement of Results

Here we state our main results and make some comments on their relationship to known results and on their proofs. Hereafter, given a function f, define f_r(x) =f(x)−rx,r∈R, andf_s(x) =f(x)−s(x), wheresis any polynomial of degree at most three such thats(0) = 0. Also, in the statements of the results, the intervalsJiand the pointsci,1≤i≤n, are defined in terms of the partition for the trapezoidal rule, and the intervalsI_i and points a_i andb_i are defined in terms of the partition for Simpson’s rule. Precise definitions are given in Section 2below.

Theorem 1.1. Letf ∈Λ_α(I),0< α≤1. Then forn≥1,

(1.4) E_n^T(f)≤ |I|^1+α

(1 +α)2^αn^αinf

r kf_rk_Λ_α,

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and

(1.5) E_2n^S (f)≤ 2(1 + 2^α+1)|I|^1+α (1 +α)6^1+αn^α inf

s kf_sk_Λ_α,

Further, inequality (1.4) is sharp, in the sense that for each n there exists a functionf such that equality holds.

Remark 1.1. We conjecture that inequality (1.5) is sharp, but we have been unable to construct an example which shows this.

Remark 1.2. Inequality (1.4) should be compared to the examples of increasing functions in Λ_α, 0 < α < 1, constructed by Dubuc and Topor [6], for which E_n^T(f) =O(1/n).

In the special case of Lipschitz functions (i.e., functions inΛ₁) Theorem1.1 can be improved.

Corollary 1.2. Letf ∈Λ₁. Then forn≥1,

|E_n^T(f)| ≤ |I|²

8n(M −m), (1.6)

|E_2n^S (f)| ≤ 5|I|²

72n (M−m), (1.7)

where M = sup_If⁰, m = inf_If⁰. Furthermore, equality holds in (1.6) if and only iff is such that its derivative is given by

(1.8) f⁰(t) = ±

n

X

i=1

M χ_J⁺

i (t) +mχ_J⁻

i (t)

! .

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Similarly, equality holds in (1.7) if and only if

(1.9) f⁰(t) = ±

n

X

i=1

mχ_I¹

i(t) +M χ_I²

i(t) +mχ_I³

i(t) +M χ_I⁴

i(t)

! .

Remark 1.3. Inequality (1.6) was first proved by Kim and Neugebauer [9] as a corollary to a theorem on integral means.

Theorem 1.3. Letf ∈CBV(I). Then forn ≥1,

(1.10) E_n^T(f)≤ |I|

2ninf

r kf_rk_BV,I, and

(1.11) E_2n^S (f)≤ |I|

3ninf

r kf_rk_BV,I.

Both inequalities are sharp, in the sense that for eachnthere exists a sequence of functions which show that the given constant is the best possible. Further, in each equality holds if and only if both sides are equal to zero.

Remark 1.4. Pölya and Szegö [12] proved an inequality analogous to (1.10) for rectangular approximations. However, they do not show that their result is sharp.

Theorem 1.4. Letf ∈W₁^p(I),1≤p≤ ∞. Then for alln ≥1, (1.12) E_n^T(f)≤ |I|^1+1/p⁰

2n(p⁰ + 1)^1/p⁰ inf

r kf_r⁰k_p,I,

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and

(1.13) E_n^S(f)≤ 2^1/p⁰(1 + 2^p⁰⁺¹)^1/p⁰|I|^1+1/p⁰ (p⁰+ 1)^1/p⁰6^1+1/p⁰n inf

s kf_s⁰k_p,I.

Inequality (1.12) is sharp, and when1< p <∞, equality holds if and only if (1.14) f⁰(t) =d₁

n

X

i=1

(t−c_i)^p⁰⁻¹χ_J⁺

i (t)−(c_i−t)^p⁰⁻¹χ_J⁻

i (t) +d₂,

whered₁, d₂ ∈ R. Similarly, inequality (1.13) is sharp, and when1 < p <∞, equality holds if and only if

(1.15) f⁰(t) =d₁

n

X

i=1

(t−a_i)^p⁰⁻¹χ_I²

i(t) + (t−b_i)^p⁰⁻¹χ_I⁴

i(t)

−(a_i−t)^p⁰⁻¹χ_I¹

i(t)−(b_i−t)^p⁰⁻¹χ_I³

i(t)

+d₂t² +d₃t+d₄,

whered_i ∈R,1≤i≤4.

Remark 1.5. When p = 1, p⁰ = ∞, and we interpret (1 +p⁰)^1/p⁰ and (1 + 2^p⁰⁺¹)^1/p⁰ in the limiting sense as equaling 1 and 2 respectively. In this case Theorem1.4is a special case of Theorem1.3since iff is absolutely continuous it is of bounded variation, andkf⁰k_1,I =kfk_BV,I.

Remark 1.6. When 1 < p < ∞ we can restate Theorem 1.4 in a form anal- ogous to Theorem 1.3. We define the space BVp of functions of bounded p-

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variation by

kfk_BV_p_,I = sup

Γ n

X

i=1

|f(x_i)−f(xi−1)|^p

|x_i −x_i−1|^p−1 <∞,

where the supremum is taken over all partitionsΓ ={x_i}ofI. Thenf ∈BV_pif and only if it is absolutely continuous andf⁰ ∈L^p(I), andkfk_BV_p_,I =||f⁰||_p,I. This characterization is due to F. Riesz; see, for example, Natanson [10].

Remark 1.7. When p = ∞, Theorem 1.4 is equivalent to Theorem 1.1 with α = 1, since f ∈ W₁^∞(I) if and only iff ∈ Λ₁(I), and kf⁰k∞,I = kfk_Λ₁_,I. (See, for example, Natanson [10].)

Remark 1.8. Inequality (1.12), withr= 0andp >1was independently proved by Dragomir [5] as a corollary to a rather lengthy general theorem. Very re- cently, we learned that Dragomir et al. [4] gave a direct proof similar to ours for (1.12) for allpbut still withr = 0. Neither paper considers the question of sharpness.

While inequalities (1.12) and (1.13) are sharp in the sense that for a givenn equality holds for a given function,E_n^T(f)andE_2n^S (f)go to zero more quickly than1/n.

Theorem 1.5. Letf ∈W₁^p(I),1≤p≤ ∞. Then

n→∞lim n·E_n^T(f) = 0 (1.16)

n→∞lim n·E_2n^S (f) = 0.

(1.17)

Further, these limits are sharp in the sense that the factor of n cannot be re- placed byn^afor anya >1.

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Remark 1.9. Unlike most of our proofs, the proof of Theorem1.5requires that we approximatef by smooth functions. It would be of interest to find a proof of this result which avoided this.

Theorem 1.6. Letf ∈W₁^pq(I),1≤p, q ≤ ∞. Then forn≥1, (1.18) E_n^T(f)≤B(q⁰/p⁰, q⁰+ 1)^1/q⁰|I|^1+1/p⁰

2n inf

r kf_r⁰k_pq,I, whereB is the Beta function,

B(u, v) = Z 1

0

x^u−1(1−x)^v−1dx, u, v >0.

Similarly,

(1.19) E_2n^S (f)≤C(q⁰/p⁰, q⁰ + 1)^1/q⁰|I|^1+1/p⁰ n inf

s kf_s⁰k_pq,I, where

C(u, v) = Z 1/3

0

t^u−1 1

3− t 2

v−1

dt+ Z 1

1/3

t^u−1 1

4− t 4

v−1

dt.

Remark 1.10. Whenp=qthen Theorem1.6reduces to Theorem1.4.

Remark 1.11. Theorem (1.6) is sharp; when 1 ≤ q < p the condition for equality to hold is straightforward (f is constant), but whenq ≥ p it is more technical, and so we defer the statement until after the proof, when we have made the requisite definitions.

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Remark 1.12. The constant in (1.19) is considerably more complicated than that in (1.18); the functionC(u, v)can be rewritten in terms of the Beta function and the hypergeometric function₂F₁, but the resulting expression is no simpler.

(Details are left to the reader.) However it is easy to show thatC(q⁰/p⁰, p⁰+1)≤ B(q⁰/p⁰, p⁰+ 1)/3^q⁰, so that we have the weaker but somewhat more tractable estimate

E_2n^S (f)≤B(q⁰/p⁰, q⁰+ 1)^1/q⁰|I|^1+1/p⁰ 3n inf

s kf_s⁰k_pq,I. Theorem 1.7. Letf ∈W₂^p(I),1≤p≤ ∞. Then forn≥1, (1.20) E_n^T(f)≤B(p⁰+ 1, p⁰+ 1)^1/p⁰|I|^2+1/p⁰

2n² kf⁰⁰k_p,I and

(1.21) E_2n^S (f)≤D(p⁰+ 1, p⁰+ 1)^1/p⁰ |I|^2+1/p⁰ 2^1/p3^2+1/p⁰n² inf

s kf_s⁰⁰kp,I, where

D(u, v) = Z 3/2

0

t^u−1|1−t|^v−1dt.

Inequality (1.20) is sharp, and when1< p <∞equality holds if and only if

(1.22) f⁰⁰(t) =d

n

X

i=1

|J_i|²

4 −(t−ci)² p⁰−1

χJi(t),

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where d ∈ R. Similarly, inequality (1.21) is sharp, and when 1 < p < ∞ equality holds if and only if

(1.23) f⁰⁰(t) =d1 n

X

i=1

|I_i|²

36 −(t−ai)² p⁰−1

χI˜_i¹(t)

−

(t−a_i)²− |I_i|² 36

p⁰−1

χI˜_i²(t)

+ |Ii|²

36 −(t−b_i)² p⁰−1

χI˜_i³(t)

−

(t−b_i)²− |I_i|² 36

p⁰−1

χI˜_i⁴(t)

!

+d₂t+d₃,

where d_i ∈ R, 1 ≤ i ≤ 3, and the intervals I˜_i^j, 1 ≤ j ≤ 4, defined in (5.2) below, are such that the corresponding functions are positive.

Remark 1.13. Whenp = 1, p⁰ = ∞, and we interpretB(p⁰+ 1, p⁰+ 1)^1/p⁰ as the limiting value 1/4. This follows immediately from the identity B(u, v) = Γ(u)Γ(v)/Γ(u+v)and from Stirling’s formula. (See, for instance, Whittaker and Watson [20].)

Remark 1.14. When p = ∞, (1.20) reduces to the classical estimate given above.

Remark 1.15. Like the function C(u, v)in Theorem 1.6, the function D(u, v) can be rewritten in terms of the Beta function and the hypergeometric function

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http://jipam.vu.edu.au 2F₁. However, the resulting expression does not seem significantly better, and

details are left to the reader.

Prior to Theorem1.7, each of our results shows that for rough functions, the trapezoidal rule is better than Simpson’s rule. More precisely, the constants in the sharp error bounds for E_2n^T (f)are less than or equal to the constants in the sharp error bounds forE_2n^S(f). (We useE_2n^T (f)instead ofE_n^T(f)since we want to compare numerical approximations with the same number of data points.)

This is no longer the case for twice differentiable functions. Numerical cal- culations show that, for instance, when p = 10/9, the constant in (1.20) is smaller, but whenp= 10, (1.21) has the smaller constant. Furthermore, the following analogue of Theorem 1.5 shows that though the constants in Theorem 1.7are sharp, Simpson’s rule is asymptotically better than the trapezoidal rule.

Theorem 1.8. Givenf ∈W₂^p(I),1≤p≤ ∞,

(1.24) lim

n→∞n²E_n^T(f) =

|I|² 12

Z

I

f⁰⁰(t)dt ,

but

(1.25) lim

n→∞n²E_2n^S (f) = 0.

Remark 1.16. (Added in proof.) Given Theorems1.5and1.8, it would be inter- esting to compare the asymptotic behavior ofE_n^T(f)andE_2n^S (f)for extremely rough functions, say those in Λα(I) and CBV(I). We suspect that in these cases their behavior is the same, but we have no evidence for this. (We want to thank the referee for raising this question with us.)

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1.3. Organization of the Paper

The remainder of this paper is organized as follows. In Section 2 we make some preliminary observations and define notation that will be used in all of our proofs. In Section 3we prove Theorems1.1and1.3 and Corollary1.2. In Section4we prove Theorems1.4,1.5and1.6. In Section5we prove Theorems 1.7and1.8.

Throughout this paper all notation is standard or will be defined when needed.

Given an intervalI,|I|will denote its length. Givenp, 1≤p ≤ ∞,p⁰ will denote the conjugate exponent: 1/p+ 1/p⁰ = 1.

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2. Preliminary Remarks

In this section we establish notation and make some observations which will be used in the subsequent proofs.

2.1. Estimating the Error

Given an interval I = [a, b], for the trapezoidal rule we will always have an equally spaced partition of n+ 1points, x_i = a+i|I|/n. Define the intervals J_i = [xi−1, x_i],1≤i≤n; then|J_i|=|I|/n.

For eachi,1≤i≤n, define (2.1) Li = |J_i|

2 f(xi−1) +f(xi)

− Z

Ji

f(t)dt.

If we divide eachJ_iinto two intervalsJ_i⁻andJ_i⁺of equal length, then (2.1) can be rewritten as

(2.2) L_i = Z

J_i⁻

f(x_i−1)−f(t) dt+

Z

J_i⁺

f(x_i)−f(t) dt.

Alternatively, if f is absolutely continuous, then we can apply integration by parts to (2.1) to get that

(2.3) L_i =

Z

Ji

(t−c_i)f⁰(t)dt,

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wherec_i = (xi−1 +x_i)/2is the midpoint ofJ_i. If f⁰ is absolutely continuous, then we can apply integration by parts again to get

(2.4) Li = 1

2 Z

Ji

|J_i|²

4 −(t−ci)²

f⁰⁰(t)dt.

From the definition of the trapezoidal rule (1.1) it follows immediately that

(2.5) E_n^T(f) =

n

X

i=1

L_i

≤

n

X

i=1

|L_i|, and our principal problem will be to estimate|L_i|.

We make similar definitions for Simpson’s rule. GivenI, we form a partition with 2n + 1 points, x_j = a+j|I|/2n, 0 ≤ j ≤ 2n, and form the intervals Ii = [x2i−2, x2i],1≤i≤n. Then|Ii|=|I|/n.

For eachi,1≤i≤n, define (2.6) K_i = |I|

6n f(x2i−2) + 4f(x2i−1) +f(x_2i)

− Z

Ii

f(t)dt.

To get an identity analogous to (2.2), we need to partitionI_i into four intervals of different lengths. Define

a_i = 2x2i−2+x2i−1

3 b_i = 2x_2i +x2i−1

3 ,

and let

I_i¹ = [x2i−2, ai], I_i² = [ai, x2i−1], I_i³ = [x2i−1, bi], I_i⁴ = [bi, x2i].

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Then|I_i¹| =|I_i⁴| =|I|/6n and|I_i²| = |I_i³| = |I|/3n, and we can rewrite (2.6) as

(2.7) K_i = Z

I_i¹

f(x2i−2)−f(t) dt+

Z

I_i²

f(x2i−1)−f(t) dt +

Z

I_i³

f(x2i−1)−f(t) dt+

Z

I_i⁴

f(x2i)−f(t) dt.

Iff is absolutely continuous we can apply integration by parts to (2.6) to get

(2.8) K_i =

Z

I_i⁻

(t−a_i)f⁰(t)dt+ Z

I⁺_i

(t−b_i)f⁰(t)dt.

Iff⁰ is absolutely continuous we can integrate by parts again to get (2.9) K_i = 1

2 Z

I_i⁻

|I_i|²

36 −(t−a_i)²

f⁰⁰(t)dt

+ 1 2

Z

I_i⁺

|Ii|²

36 −(t−b_i)²

f⁰⁰(t)dt.

Whichever expression we use, it follows from the definition of Simpson’s rule (1.2) that

(2.10) E_2n^S (f) =

n

X

i=1

K_i

≤

n

X

i=1

|K_i|.

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Finally, we want to note that there is a connection between Simpson’s rule and the trapezoidal rule: it follows from the definitions (1.1) and (1.2) that

(2.11) S2n(f) = 4

3T2n(f)− 1 3Tn(f).

2.2. Modifying the Norm

In all of our results, we estimate the error in the trapezoidal rule with an expression of the form

infr kf_rk,

where the infimum is taken over allr∈R. It will be enough to prove the various inequalities with kfkon the righthand side: since the trapezoidal rule is exact on linear functions,E_n^T(f_r) = E_n^T(f)for allf andr. Further, we note that for eachf, there existsr0 ∈Rsuch that

kf_r₀k= inf

r kf_rk.

This follows since the norm is continuous inrand tends to infinity as|r| → ∞.

Similarly, in our estimates forE_2n^S(f), it will suffice to prove the inequalities with kfk on the righthand side instead of inf_skf_sk: because Simpson’s rule is exact for polynomials of degree 3 or less, E_n^S(f) = E_n^T(f_s), for s(x) = ax³+bx²+cx. Again the infimum is attained, since the norm is continuous in the coefficients ofsand tends to infinity as|a|+|b|+|c| → ∞.

(The exactness of the Trapezoidal rule and Simpson’s rule is well-known;

see, for example, Ralston [13].)

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3. Functions in Λ

_α

(I), 0 < α ≤ 1, and CBV (I )

Proof of Theorem1.1. We first prove inequality (1.4). By (2.2), for eachi,1≤ i≤n,

|L_i| ≤ Z

J_i⁻

|f(xi−1)−f(t)|dt+ Z

J_i⁺

|f(x_i)−f(t)|dt

≤ kfk_Λ_α Z

J_i⁻

|x_i−1−t|^αdt+kfk_Λ_α Z

J_i⁻

|x_i−t|^αdt;

by translation and reflection,

= 2kfk_Λ_α Z

J_i⁻

(t−xi−1)^αdt

= 2kfk_Λ_α|J_i⁻|^1+α 1 +α

= |I|^1+αkfkΛα

(1 +α)2^αn^1+α. Therefore, by (2.5)

(3.1) E_n^T(f)≤ |I|^1+αkfkΛα

(1 +α)2^αn^α, and by the observation in Section2.2we get (1.4).

The proof of inequality (1.5) is almost identical to the proof of (1.4): we begin with inequality (2.7) and argue as before to get

|K_i| ≤ 2(1 + 2^α+1)|I|^1+αkfk_Λ_α (1 +α)6^1+αn^1+α ,

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which in turn implies (1.5).

To see that inequality (1.4) is sharp, fixn ≥ 1and define the function f as follows: on[0,1/n]let

f(x) =











x^α, 0≤x≤ 1 2n

x− 1 n

α

, 1

2n ≤x≤ 1 n.

Now extendf to the interval[0,1]as a periodic function with period1/n. It is clear that kfk_Λ_α = 1, and it is immediate from the definition thatT_n(f) = 0.

Therefore,

E_n^T(f) = Z 1

0

f(x)dx= 2n Z 1/2n

0

x^αdx= 1 (1 +α)2^αn^α, which is precisely the righthand side of (3.1).

Proof of Corollary1.2. Inequalities (1.6) and (1.7) follow immediately from (1.4) and (1.5). Recall that if f ∈ Λ₁(I), then f is differentiable almost ev- erywhere,f⁰ ∈L^∞(I)andkfk_Λ₁ =kf⁰k∞. (See, for example, Natanson [10].) Letr= (M+m)/2; then

kf −rxk_Λ₁ =kf⁰−rk∞= M−m 2 .

We now show that (1.6) is sharp and that equality holds exactly when (1.8) holds. First note that if (1.8) holds, then by (2.3),

L_i = Z

J_i⁺

(t−c_i)M dt+ Z

J_i⁻

(t−c_i)m dt=±|I|²

8n²(M −m),

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and it follows at once from (2.5) that equality holds in (1.6).

To prove that (1.8) is necessary for (1.6) to hold, we consider two cases.

Case 3.1. M > 0andm =−M. In this case, E_n^T(f) = |I|²

4nM.

Again by (2.3), E_n^T(f)≤

n

X

i=1

Z

Ji

(t−c_i)f⁰(t)dt

≤

n

X

i=1

Z

J_i⁻

(t−c_i)f⁰(t)dt

+ Z

J_i⁺

(t−c_i)f⁰(t)dt

≤ |I|² 8n²

n

X

i=1

kf⁰k_∞,J⁻

i +kf⁰k_∞,J⁺

i

≤ |I|²

8n M +|I|² 8n M,

and since the first and last terms are equal, equality must hold throughout.

Therefore, we must have that (3.2) |L⁻_i |=kf⁰k_∞,J⁻

i

Z

J_i⁻

(c_i−t)dt, |L⁺_i |=kf⁰k_∞,J⁺

i

Z

J_i⁺

(t−c_i)dt, and

(3.3) |I|²

8n²

n

X

i=1

kf⁰k_∞,J⁺

i = |I|² 8n²

n

X

i=1

kf⁰k_∞,J⁻

i = |I|² 8n M.

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Hence, by (3.2), onJ_i

f⁰(t) =α_iχ_J⁺

i (t)−β_iχ_J⁻

i (t),

with eitherα_i, β_i >0for alli, orα_i, β_i <0for alli. Without loss of generality we assume thatα_i, β_i >0.

Further, we must have thatM = sup{α_i : 1 ≤ i ≤ n}, so it follows from (3.3) thatα_i = M for alli. Similarly, we must have thatβ_i = M,1 ≤ i ≤ n.

This completes the proof of Case3.1.

Case 3.2. The general case: m < M. Letr= (M +m)/2; then

E_n^T(f) = |I|²

8n (M −m) =E_n^T(f_r).

Since Case3.1applies tof_r, we have that

f_r⁰(t) = M −m 2

n

X

i=1

χ_J⁺

i (t)−χ_J⁻

i (t) .

This completes the proof sincef⁰ =f_r⁰ +r.

The proof that (1.7) is sharp and equality holds if and only if (1.9) holds is essentially the same as the above argument, and we omit the details.

Proof of Theorem1.3. We first prove (1.10). By (2.2) and the definition of the norm inCBV(I), for eachi,1≤i≤n,

|L_i| ≤ Z

J_i⁻

|f(xi−1)−f(t)|dt+ Z

J_i⁺

|f(x_i)−f(t)|dt

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≤ kfk_BV,J⁻

i |J_i⁻|+kfk_BV,J⁺

i |J_i⁺|

= 1

2nkfk_BV,J_i. If we sum overi, we get

E_n^T(f)≤ 1 2n

n

X

i=1

kfk_BV,J_i = 1

2nkfk_BV,I; inequality (1.10) now follows from the remark in Section2.2.

To show that inequality (1.10) is sharp, fix n ≥ 1 and for k ≥ 1 define a_k = 4^−k/n. We now define the function f_k on I = [0,1] as follows: on [0,1/n]let

f_k(x) =











1− x

a_n 0≤x≤a_n

0 a_n ≤x≤ 1

n −a_n

1 + x−_n¹ an

1

n −a_n≤x≤ 1 n.

Extend f_k to [0,1] periodically with period 1/n. It follows at once from the definition thatkf_kk_BV,[0,1] = 2n. Furthermore,

E_n^T(fk) =

Tn(fk)− Z 1

0

fk(t)dt

= 1−akn= 1−4^−k. Thus the constant1/2nin (1.10) is the best possible.

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We now consider when equality can hold in (1.10). Iff(t) = mt+b, then we have equality since both sides are zero.

For the converse implication we first show that iff ∈ CBV(I)is not constant onI, then

(3.4) E_n^T(f)< |I|

2nkfk_BV,I.

By the above argument, it will suffice to show that for somei,

|L_i|< |I|

2nkfk_BV,J_i.

Since f is non-constant, choose i such that f is not constant on J_i. Since f is continuous, the function|f(xi−1) +f(xi)−2f(t)|achieves its maximum at some t ∈ J_i, and, again becausef is non-constant, it must be strictly smaller than its maximum on a set of positive measure. Hence on a set of positive measure,|f(xi−1) +f(xi)−2f(t)|<kfkBV,Ji, and so (3.4) follows from (2.1), since we can rewrite this as

L_i = 1 2

Z

Ji

f(xi−1) +f(x_i)−2f(t) dt.

To finish the proof, note that as we observed in Section2.2, there existsr₀ such thatkf_r₀k_BV,I = inf_rkf_rk_BV,I. Hence, we would have that

E_n^T(f) =E_n^T(f_r₀)≤ |I|

2nkf_r₀k_BV,I.

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Iff(t)were not of the formmt+b, so thatf_r₀ could not be a constant function, then by (3.4), the inequality would be strict. Hence equality can only hold iff is linear.

The proof that inequality (1.11) holds is almost identical to the proof of (1.10): we begin with inequality (2.7) and argue exactly as we did above.

The proof that inequality (1.11) is sharp requires a small modification to the example given above. Fix n ≥ 1 and, as before, let a_k = 4^−k/n. Define the functionf_konI = [0,1]as follows: on[0,1/n]let

f_k(x) =











0 0≤x≤ 1

2n −an

1 + x− _2n¹ a_n

1

2n −a_n≤x≤ 1 2n

1 +

1 2n −x

an

1

2n ≤x≤ 1 2n +a_n

0 1

2n +an ≤x≤ 1 n.

Extendf_kto[0,1]periodically with period1/n. Then we again havekf_kk_BV,[0,1] = 2n; furthermore,

E_2n^S (f_k) =

S_2n(f_k)− Z 1

0

f_k(t)dt

= 2

3 −a_kn = 2

3 −4^−k. Thus the constant1/3nin (1.11) is the best possible.

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The proof that equality holds in (1.11) only when both sides are zero is again very similar to the above argument, replacingLi byKi and using (2.6) instead of (2.1).

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4. Functions in W

₁^p

(I ) and W

₁^pq

(I ), 1 ≤ p, q ≤ ∞

Proof of Theorem1.4. As we noted in Remarks 1.5 and1.7, it suffices to consider the case1< p <∞.

We first prove inequality (1.12). If we apply Hölder’s inequality to (2.3), then for alli,1≤i≤n,

|L_i| ≤ kf⁰k_p,J_i Z

Ji

|t−c_i|^p⁰dt 1/p⁰

.

An elementary calculation shows that Z

Ji

|t−c_i|^p⁰dt= |J_i|^p⁰⁺¹ (p⁰ + 1)2^p⁰.

Hence, by (2.5) and by Hölder’s inequality for series, E_n^T(f)≤ |I|^1+1/p⁰

2(p⁰+ 1^1/p⁰)n^1+1/p⁰

n

X

i=1

Z

Ji

|f⁰(t)|^pdt 1/p

≤ |I|^1+1/p⁰ 2(p⁰+ 1)^1/p⁰n^1+1/p⁰

Z

I

|f⁰(t)|^pdt 1/p

n^1/p⁰

= |I|^1+1/p⁰

2(p⁰ + 1)^1/p⁰nkf⁰k_p,I.

Inequality (1.12) now follows from the observation in Section2.2.